Boundary Conditions at Infinity for a Single Blocking Electrode

1964 ◽  
Vol 41 (5) ◽  
pp. 1505-1506 ◽  
Author(s):  
Ian R. Gatland ◽  
Louis Gold ◽  
John W. Moffat
Keyword(s):  
Boundary Conditions ◽  
Conditions At Infinity

Proper Boundary Conditions for Infinitely Layered Orthotropic Media

2000 ◽  
Vol 67 (3) ◽  
pp. 629-632
Author(s):  
E. L. Bonnaud ◽  
J. M. Neumeister
Keyword(s):  
Boundary Conditions ◽  
Layered Medium ◽  
Integral Transforms ◽  
Numerical Treatment ◽  
Problem Size ◽  
Transfer Matrix Approach ◽  
Stress Functions ◽  
Orthotropic Media ◽  
First Time ◽  
Conditions At Infinity

A stress analysis of a plane infinitely layered medium subjected to surface loadings is performed using Airy stress functions, integral transforms, and a revised transfer matrix approach. Proper boundary conditions at infinity are for the first time established, which reduces the problem size by one half. Methods and approximations are also presented to enable numerical treatment and to overcome difficulties inherent to such formulations. [S0021-8936(00)01103-X]

On the theory of elastic collisions between electrons and hydrogen atoms

1960 ◽  
Vol 254 (1277) ◽  
pp. 259-272 ◽  
Keyword(s):  
Wave Function ◽  
Hydrogen Atom ◽  
Boundary Conditions ◽  
Continuous Spectrum ◽  
Ground State ◽  
Wave Functions ◽  
Hydrogen Atoms ◽  
Complete Set ◽  
Exchange Scattering ◽  
Conditions At Infinity

A hydrogen atom in the ground state scatters an electron with kinetic energy too small for inelastic collisions to occur. The wave function Ψ(r 1 ; r 2 ) of the system has boundary conditions at infinity which must be chosen to allow correctly for the possibilities of both direct and exchange scattering. The expansion Ψ = Σ ψ,(r 1 )F y (r 2 ) of the total wave function in y terms of a complete set of hydrogen atom wave functions ψ y (r 1 ) includes an integration over the continuous spectrum. It is si own that the integrand contains a singularity. The explicit form of this singularity and its connexion with the boundary conditions are examined in detail. The symmetrized functions Y* may be represented by expansions of the form Σ {ψ y (r 1 ) G y ±(r 2 ) ±ψ y (r 2 ) y G y ±(r 1 )}, where the integrand in the continuous spectrum does not involve singularities. Finally, it is shown that because all the states ψ y of the hydrogen atom are included in the expansion, the equation satisfied by F 1 , the coefficient of the ground state, contains a polarization potential which behaves like — a/2 r 4 for large r and is independent of the velocity of the incident electron.

scholarly journals A New Approach for a Class of the Blasius Problem via a Transformation and Adomian’s Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdelhalim Ebaid ◽  
Nwaf Al-Armani
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Padé Approximation ◽  
Current Approach ◽  
Pade Approximation ◽  
Flow Problems ◽  
Blasius Problem ◽  
Classical Boundary ◽  
Layer Flow ◽  
Conditions At Infinity

The main feature of the boundary layer flow problems is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such problems. To the best of the authors’ knowledge, two procedures were used extensively in the past two decades to deal with the boundary conditions at infinity, either the Padé approximation or the direct numerical codes. However, an intensive work is needed to perform the calculations using the Padé technique. Regarding this point, a new idea is proposed in this paper. The idea is based on transforming the unbounded domain into a bounded one by the help of a transformation. Accordingly, the original differential equation is transformed into a singular differential equation with classical boundary conditions. The current approach is applied to solve a class of the Blasius problem and a special class of the Falkner-Skan problem via an improved version of Adomian’s method (Ebaid, 2011). In addition, the numerical results obtained by using the proposed technique are compared with the other published solutions, where good agreement has been achieved. The main characteristic of the present approach is the avoidance of the Padé approximation to deal with the infinity boundary conditions.

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